Appendices
Selina Lu
Figure 1: Summary Ouput - Initial Model
##
## Call:
## lm(formula = cmRate ~ region + pctPrivateHC + pctEmployerHC +
## pctPublicHC, data = mortalityRates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -75.968 -12.971 0.096 12.430 116.450
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 150.4018 21.6748 6.939 1.16e-11 ***
## regionNortheast -3.9740 3.6837 -1.079 0.281171
## regionSoutheast 10.7338 2.8554 3.759 0.000189 ***
## regionSouthwest -7.6100 4.3959 -1.731 0.084008 .
## regionWest -18.9862 3.7267 -5.095 4.87e-07 ***
## pctPrivateHC -0.7575 0.2080 -3.642 0.000297 ***
## pctEmployerHC 0.8252 0.2220 3.717 0.000223 ***
## pctPublicHC 1.1768 0.2567 4.584 5.69e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.08 on 529 degrees of freedom
## Multiple R-squared: 0.2955, Adjusted R-squared: 0.2861
## F-statistic: 31.69 on 7 and 529 DF, p-value: < 2.2e-16Figure 2: Residuals Plot - Initial Model
Figure 3: QQ Plot of Residuals - Initial Model
Figure 4: Scatterplot Matrix of Data
Check for Multicollinearity and Overfitting
## pctPrivateHC pctEmployerHC pctPublicHC
## 3.306927 4.060143 3.002154##
## Midwest Northeast Southeast Southwest West
## 178 53 199 51 56Figure 5: Histogram of pctPublicHC (with and without a log transformation)
Figure 5.1: Residuals of Log Transformed Reponse vs Public
Figure 5.2: Histogram of pctPrivateHC (with and without a log transformation)
Figure 5.3: Residuals of Log Transformed Reponse vs Private
Figure 6: Second Model - Removing pctEmployerHC variable to address multicollinearity
##
## Call:
## lm(formula = cmRate ~ region + pctPrivateHC + pctPublicHC, data = mortalityRates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -80.157 -13.675 0.516 13.226 113.151
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 181.7442 20.2077 8.994 < 2e-16 ***
## regionNortheast -1.3370 3.6583 -0.365 0.714906
## regionSoutheast 10.7951 2.8896 3.736 0.000207 ***
## regionSouthwest -9.6164 4.4151 -2.178 0.029841 *
## regionWest -20.9214 3.7345 -5.602 3.41e-08 ***
## pctPrivateHC -0.4339 0.1911 -2.270 0.023626 *
## pctPublicHC 0.6852 0.2227 3.077 0.002198 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.36 on 530 degrees of freedom
## Multiple R-squared: 0.2771, Adjusted R-squared: 0.2689
## F-statistic: 33.85 on 6 and 530 DF, p-value: < 2.2e-16Checking for Multicollinearity for Second Linear Model
## pctPrivateHC pctPublicHC
## 2.27376 2.27376Figure 7: Residuals and QQ Plot for Second Linear Model
***
Figure 8: Exploring Additional Variables
Figure 9: Scatterplot Matrix of Data from Additional Variable Exploration with log transformations to Average Household Size and Median Income
Figure 10: Residuals and QQ Plot of Third Linear Model with inclusion of Median Income variable
##
## Call:
## lm(formula = cmRate ~ region + pctPrivateHC + pctPublicHC + log(medianIncome),
## data = mortalityRates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -82.772 -13.206 0.423 13.530 110.317
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 357.2177 92.9614 3.843 0.000136 ***
## regionNortheast 1.1095 3.8620 0.287 0.774005
## regionSoutheast 10.9771 2.8837 3.807 0.000157 ***
## regionSouthwest -8.1766 4.4663 -1.831 0.067701 .
## regionWest -19.0671 3.8464 -4.957 9.65e-07 ***
## pctPrivateHC -0.2557 0.2118 -1.207 0.227814
## pctPublicHC 0.4747 0.2473 1.919 0.055480 .
## log(medianIncome) -16.7770 8.6766 -1.934 0.053697 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.3 on 529 degrees of freedom
## Multiple R-squared: 0.2821, Adjusted R-squared: 0.2726
## F-statistic: 29.7 on 7 and 529 DF, p-value: < 2.2e-16
Check for Multicollinearity
vif(lm(cmRate ~ pctPrivateHC + pctPublicHC + log(medianIncome), data = mortalityRates))## pctPrivateHC pctPublicHC log(medianIncome)
## 2.947409 2.935322 3.334527Figure 11: Testing for Interactions - Scatterplot of pctPrivateHC vs cmRate by Region
***
Figure 12: Model with Interaction for Plot
##
## Call:
## lm(formula = cmRate ~ pctPrivateHC * region, data = mortalityRates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -82.646 -12.910 1.238 13.341 108.228
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 273.8904 17.9985 15.217 < 2e-16 ***
## pctPrivateHC -1.3981 0.2514 -5.562 4.25e-08 ***
## regionNortheast -9.5693 44.5894 -0.215 0.83015
## regionSoutheast -31.9221 20.5643 -1.552 0.12119
## regionSouthwest -87.3600 27.9584 -3.125 0.00188 **
## regionWest -78.8535 32.2949 -2.442 0.01495 *
## pctPrivateHC:regionNortheast 0.1197 0.6155 0.194 0.84587
## pctPrivateHC:regionSoutheast 0.5791 0.3006 1.927 0.05456 .
## pctPrivateHC:regionSouthwest 1.1228 0.4405 2.549 0.01110 *
## pctPrivateHC:regionWest 0.8193 0.4866 1.684 0.09283 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.45 on 527 degrees of freedom
## Multiple R-squared: 0.2754, Adjusted R-squared: 0.263
## F-statistic: 22.26 on 9 and 527 DF, p-value: < 2.2e-16Figure 13: Fourth Linear Model Comparison without and with Interaction
##
## Call:
## lm(formula = cmRate ~ log(medianIncome) + pctPublicHC + pctPrivateHC +
## region, data = mortalityRates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -82.772 -13.206 0.423 13.530 110.317
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 357.2177 92.9614 3.843 0.000136 ***
## log(medianIncome) -16.7770 8.6766 -1.934 0.053697 .
## pctPublicHC 0.4747 0.2473 1.919 0.055480 .
## pctPrivateHC -0.2557 0.2118 -1.207 0.227814
## regionNortheast 1.1095 3.8620 0.287 0.774005
## regionSoutheast 10.9771 2.8837 3.807 0.000157 ***
## regionSouthwest -8.1766 4.4663 -1.831 0.067701 .
## regionWest -19.0671 3.8464 -4.957 9.65e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.3 on 529 degrees of freedom
## Multiple R-squared: 0.2821, Adjusted R-squared: 0.2726
## F-statistic: 29.7 on 7 and 529 DF, p-value: < 2.2e-16##
## Call:
## lm(formula = cmRate ~ log(medianIncome) + pctPublicHC + pctPrivateHC *
## region, data = mortalityRates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -81.083 -13.557 1.003 12.978 105.507
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 399.1541 97.4688 4.095 4.88e-05 ***
## log(medianIncome) -17.0372 8.9014 -1.914 0.05616 .
## pctPublicHC 0.5051 0.2514 2.009 0.04502 *
## pctPrivateHC -0.8194 0.2929 -2.797 0.00534 **
## regionNortheast -37.4411 44.7276 -0.837 0.40292
## regionSoutheast -36.3780 20.4196 -1.782 0.07541 .
## regionSouthwest -82.6308 27.6938 -2.984 0.00298 **
## regionWest -80.6499 32.5117 -2.481 0.01343 *
## pctPrivateHC:regionNortheast 0.5424 0.6196 0.875 0.38180
## pctPrivateHC:regionSoutheast 0.6814 0.2990 2.279 0.02307 *
## pctPrivateHC:regionSouthwest 1.1500 0.4358 2.639 0.00857 **
## pctPrivateHC:regionWest 0.8977 0.4895 1.834 0.06723 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.19 on 525 degrees of freedom
## Multiple R-squared: 0.2941, Adjusted R-squared: 0.2794
## F-statistic: 19.89 on 11 and 525 DF, p-value: < 2.2e-16Figure 14: Residuals and QQ Plot of Fourth Linear Model with Interaction
***
Checking for Multicollinearity of Fourth Model
## log(medianIncome) pctPublicHC pctPrivateHC
## 3.334527 2.935322 2.947409Figure 15: Fifth Linear Model Comparison with no Region
##
## Call:
## lm(formula = cmRate ~ pctPrivateHC + pctPublicHC + log(medianIncome),
## data = mortalityRates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -89.154 -14.268 1.397 15.653 104.967
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 539.5016 91.1376 5.920 5.78e-09 ***
## pctPrivateHC -0.1526 0.1803 -0.846 0.398
## pctPublicHC 0.3607 0.2438 1.479 0.140
## log(medianIncome) -33.8663 8.4751 -3.996 7.35e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 24.89 on 533 degrees of freedom
## Multiple R-squared: 0.1747, Adjusted R-squared: 0.17
## F-statistic: 37.6 on 3 and 533 DF, p-value: < 2.2e-16Figure 16: Final Model - CMF
\(E(log(Y_{cmRate})|X) = \beta_0 + \beta_1 log(X_{medianIncome}) + \beta_2 X_{pctPublicHC} + \beta_3 X_{pctPrivateHC} + \beta_4 I_{NE} + \beta_5 I_{SE} + \beta_6 I_{SW} + \beta_7 I_{W}\) \(+ \beta_8X_{pctPrivHC} I_{NE} + \beta_{9}X_{pctPrivHC} I_{SE} + \beta_{10}X_{pctPrivHC} I_{SW} + \beta_{11}X_{pctPrivHC} I_{W}\)
Figure 17: Final Model
##
## Call:
## lm(formula = cmRate ~ log(medianIncome) + pctPublicHC + pctPrivateHC *
## region, data = mortalityRates)
##
## Residuals:
## Min 1Q Median 3Q Max
## -81.083 -13.557 1.003 12.978 105.507
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 399.1541 97.4688 4.095 4.88e-05 ***
## log(medianIncome) -17.0372 8.9014 -1.914 0.05616 .
## pctPublicHC 0.5051 0.2514 2.009 0.04502 *
## pctPrivateHC -0.8194 0.2929 -2.797 0.00534 **
## regionNortheast -37.4411 44.7276 -0.837 0.40292
## regionSoutheast -36.3780 20.4196 -1.782 0.07541 .
## regionSouthwest -82.6308 27.6938 -2.984 0.00298 **
## regionWest -80.6499 32.5117 -2.481 0.01343 *
## pctPrivateHC:regionNortheast 0.5424 0.6196 0.875 0.38180
## pctPrivateHC:regionSoutheast 0.6814 0.2990 2.279 0.02307 *
## pctPrivateHC:regionSouthwest 1.1500 0.4358 2.639 0.00857 **
## pctPrivateHC:regionWest 0.8977 0.4895 1.834 0.06723 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.19 on 525 degrees of freedom
## Multiple R-squared: 0.2941, Adjusted R-squared: 0.2794
## F-statistic: 19.89 on 11 and 525 DF, p-value: < 2.2e-16
## pctPrivateHC pctPublicHC log(medianIncome)
## 2.947409 2.935322 3.334527